# Utility Maximization under a Budget Constraint

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```							ECON 326, University of Maryland
Instructor: Vesela Grozeva, www.econ.umd.edu/~grozeva

Example of Constrained Utility Maximization

Suppose that you have a fixed income of \$400 and you consume only two goods: food
and gas. The price of a gallon of gas, Pg, equals \$2 and the price of one unit of food, Pf,
equals \$5. Further assume that you spend your entire income on consumption only (i.e.
you do not save).

Your utility function has the following functional form:

U(gas, food) = (F)1/2(G)1/2     (the square root of the product of food and gas)

a. Write down the equation of your budget constraint. If gas is plotted on the vertical axis
and food – on the horizontal axis, what is the slope of the budget line? What is the
maximum amount of gas you can afford?

b. Compute the marginal rate of substitution as a function of gas and shelter.

c. Draw the budget constraint and an indifference curve characterized by the utility
function above. Graphically, show how the optimal consumption bundle is determined.

d. Algebraically, solve for the optimal levels of consumption of the two goods.

(The solution follows on the next page)

1
ECON 326, University of Maryland
Instructor: Vesela Grozeva, www.econ.umd.edu/~grozeva

Solution:

a.      I = Pf * F + Pg * G

400 = 5F + 2G
or
G = 200 – (5/2)F                    (Budget equation in slope-intercept form)

Slope = -5/2 = -2.5

Maximum amount of gas = vertical intercept = I/Pg = 400/2 = 200 gallons

b.      MRS = MUf / MUg where MU stands for marginal utility

MUf = dU/dF = (1/2) F -1/2 G 1/2 = .5 (G/F) 1/2

MUg = dU/dG = (1/2) F 1/2 G -1/2 = .5 (F/G) 1/2

(if you do not understand the operations above, please see the appendix at the end)

MRS = [.5 F -1/2 G 1/2 ] / [.5 F 1/2 G -1/2 ] = G / F

MRS = G / F

The marginal rate of substitution characterizes the slope of the indifference curve.
Its value depends on where on the curve we are and therefore MRS is not constant but a
function of the consumption bundle we are looking at. We know that the typical
indifference curve has a negative slope and we are only interested in the absolute value of
the MRS, denoted as | MRS |.
For example, if you are consuming 100 gallons of gas and 80 units of food, the MRS at
the point (80,100) equals 100/80, or 1.25. This value can be interpreted in the following
way: if you are currently buying 100 gallons of gas and 80 units of food, you are willing
to give up 1.25 gallons of gas to obtain one more unit of food.

2
ECON 326, University of Maryland
Instructor: Vesela Grozeva, www.econ.umd.edu/~grozeva

c.
Gas
(gallons)
200

(F*, G*)
G*

IC

Budget Line

F*                                      Food
80

d. Maximization rule:           Pf / Pg = MRS

5/2=G/F

G* = 2.5 F*
Note that the maximization rule does not yield the specific amounts of food and gas that
will maximize utility subject to the budget constraint. Instead, the maximization rule only
tells us what the optimal ratio of food and gas is. That is, we know that if we are to
optimize, we should always purchase food and gas in the indicated proportion (we need
to buy 2.5 times as much gas as food where this proportion depends on the units of
measurement).
To find the exact amounts of food and gas that we need to purchase we need to take into
account the information in the budget constraint. Also, notice that the equation above has
two unknowns so that we can express G as a function of F (or the other way around) and
then use the budget constraint equation to find F:

G = 2.5 F

Now, plug in this expression for G into the BC:
200 = G + 2.5F
G = 200 – 2.5 F
2.5 F = 200 – 2.5 F
+ 2.5 F       + 2.5F

3
ECON 326, University of Maryland
Instructor: Vesela Grozeva, www.econ.umd.edu/~grozeva

5 F = 200
F = 200/5
F* = 40 units
G* = 2.5 F* = (2.5)(40) = 100 gallons

Appendix

Finding the marginal utility of good X involves taking the partial derivative of the utility
function with respect to X and treating Y as constant.
Here is the only differentiation rule that we will use for this class:

Y = f(X) = cXn ,                c - constant

dY/dX = c*n*Xn-1

Examples:
Function of X and Y                         Its derivative with respect to X

X3                                          3X3-1 = 3X2

2X4                                        (2)(4)X3 = 8X3

X1/2                                          (1/2)X-1/2

5X                                               5

XY                                               Y

4X2Y3                                     [(4)(2)X]Y3 = 8XY3

X2/3Y1/3                                      (2/3)X-1/3 Y1/3

4
ECON 326, University of Maryland
Instructor: Vesela Grozeva, www.econ.umd.edu/~grozeva

1 (or any number)                         0

5

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